Excel
02 Dec 2008
Apr
17
Historical parametric approaches to volatility (VaR)
by David Harper, CFA, FRM, CIPM
In a previous post, we sketched the ways to estimate current volatility. We now rephrase: how can we estimate of value at risk (VaR)? But we will only here talk about historically-informed approaches, not forward-looking Monte Carlo methods.
Volatility is a parameter. Return VaR (%) is just a multiple of volatility
In the most common parametric approach, there is a clean linear relationship between asset volatility and value-at-risk (VaR). VaR is just a multiple of volatility. The multiple is a based on our desired confidence (e.g., 95% or 99% confidence levels) and the time horizon: the greater our sought confidence and the longer the time horizon, the more we have to lose.
Put another way, the single-period VaR with 84.1% confidence is exactly equal to volatility. That’s because 84.1% of a cumulative normal distribution is “to the right of” negative one standard deviation (-1 SD). So, over a single period, we expect with 84.1% confidence to lose no more than one standard deviation. Starting from here, as we increase confidence and/or extend into multiple periods, we increase the VaR as a multiple of the volatility. Of course, we could multiply by portfolio value, and instead express VaR as a dollar value ($) instead of a percentage return (%) But no matter, it is still the case: VaR scales with, is a multiple of, volatility.
Parametric versus Simple Historical Simulation (line ‘em up, no volatility parameter needed)
But the above is a parametric approach: it imposes or assumes a parametric distribution, described by parameters such as standard deviation, on the pattern of asset returns. In regard to historically-based approaches, we can employ either a non-parametric approach or a parametric approach:
The basic non-parametric approach to VaR is historical simulation. This way doesn’t really compute a volatility (by definition: no volatility = no parameter = non-parametric). Historical simulation simply collects a set of historical returns and order them from top to bottom (a confusion is created by the occasional use of the term ‘non-parametric volatility.’ When they say this, they really should say non-parametric VaR).
Unweighted (STDEV), GARCH, and EWMA (incl. RM)
In the realm of historical, parametric value-at-risk (VaR), the basic idea is to assume normal distribution of asset returns. As you may know, this is both natural and totally wrong. Asset returns tend to show fat not normal tails. But the normal distribution is utterly convenient.
In the introductory FRM readings (Allen and Hull), we are introduced to at least four historical, parametric approaches (not including hybrids). Don’t lose sight: each of the four below are just different ways to estimate volatility as the “parameter” input into the VaR estimate!
The unweighted approach (Linda Allen calls this STDEV) simply computes the variance from a historical sample of returns. We already showed how EWMA is a special case of GARCH(1,1). And, finally, RiskMetricsTM is branded flavor of EMWA. RiskMetrics is EWMA where the lambda is set to about 94%; lambda is the rate of decay/persistence. You need to know the pros and cons of each, which include
|
Approach |
Pros |
Cons |
|---|---|---|
| STDEV | Simple, common | Ignores order, ghosting |
| EMWA & RM | Recursive, empahsizes recent | Ignores mean reversion |
| GARCH(1,1) | Accurate, flexible | Occassionally inappropriate |
Comments
David,
I understand the 3 broad approaches to VaR:
1. Non-parametric (historical simulation)
2. Market Implied
3. Parametric
A. Unweighted: STDEV
B. Weighted: EWMA (RM), GARCH (1,1)
In this “Approaches to VaR” organization, where does the Monte Carlo method and VCV (variance-covariance, or delta-normal) model fit in?
Thanks,
J.W.
Hi J.W.,
I realize L. Allen includes market implied as a VaR approach, but it seems confusing (the volatility is implied, not the VaR per se), so I really think the three basic approaches are:
1. Historical simulation (non-param)
2. Parametric
3. Monte Carlo
Or, the way I simplistically remember them, one looks back with data, another looks forward with fake data (MCS) and the parametric avoids data.
That scheme follow Jorion, where he just tends to call the parametric delta-normal. But parametric is a broad umbrella with many sub-classes including delta-normal. Sure delta normal is also called variance-covariance but I sort of don’t like that label here b/c it has an overlapping but broader meaning (doesn’t it imply a portfolio?)...but IMO if we are talking about VaR approaches (and not volatility), then i would take your list and: replace market implied with Monte Carlo and then just slot VCV as a specific sub-class under parametric (where we have hardly been comprehensive anyhow)
I hope that helps....
a learner visit here so as to have more knowledge of risk management.
Thank you!
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